This answer includes the problem:
Suppose you have a 2-d bullet going very fast through a 2-d gas. The gas molecules reflects specularly off the bullet, making glancing collisions. What shape of bullet of a fixed area has the least drag?
This problem gives
$$\int {1\over 1+y'^2} + \lambda y dx $$
And the equation for y' you get is
$$ y' = \lambda x (1 - 2 y'^2 - y'^4) $$
or
$$ y = \int \lambda x ( 1 - 2 y^2 - y^4) $$
Which you can solve in a series by plugging in $y={\lambda x^2\over 2}$ and iterating a few times using the relation above as a recursion.
I am a beginner in calculus of variations and I can not understand the solution in that answer.
Is there any extra assumption in the problem's statement? What should the exact statement of the underlying physics/mathematics model be? I have difficulties in writing down the optimization equations of "has the least drag"
